Cartesian 3-D - Maple Help

Average Value of a Function: $f\left(x,y,z\right)$

 Description In the Cartesian coordinate system, determine the average value of a multivariate function.

Average Value of a Function: $f\left(x,y,z\right)$

Function

 > ${{x}}^{{3}}{{y}}^{{2}}{z}$
 ${{x}}^{{3}}{}{{y}}^{{2}}{}{z}$ (1)

Region: $\left\{{z}_{1}\left(x,y\right)\le z\le {z}_{2}\left(x,y\right),{y}_{1}\left(x\right)\le y\le {y}_{2}\left(x\right),a\le x\le b\right\}$

${z}_{1}\left(x,y\right)$

 > ${1}{-}{x}{-}{y}$
 ${1}{-}{x}{-}{y}$ (2)

${z}_{2}\left(x,y\right)$

 > ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$
 ${10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}$ (3)

${y}_{1}\left(x\right)$

 > ${{x}}^{{2}}$
 ${{x}}^{{2}}$ (4)

${y}_{2}\left(x\right)$

 > ${x}$
 ${x}$ (5)

$a$

 > ${0}$
 ${0}$ (6)

$b$

 > ${1}$
 ${1}$ (7)

Inert integral:

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{z}=..,{y}=..,{x}=..,\mathrm{output}=\mathrm{integral}\right)$
 $\frac{{140}}{{219}}{}{{∫}}_{{0}}^{{1}}{{∫}}_{{{x}}^{{2}}}^{{x}}{{∫}}_{{1}{-}{x}{-}{y}}^{{10}{-}{{x}}^{{2}}{-}{{y}}^{{2}}}{{x}}^{{3}}{}{{y}}^{{2}}{}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{z}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}$ (8)

Value

 > $\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\left[\mathrm{FunctionAverage}\right]\left(,{z}=..,{y}=..,{x}=..\right)$
 $\frac{{72239}}{{204984}}$ (9)

 Commands Used